General Relativity. on the frame of reference!

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1 General Relativity Problems with special relativity What makes inertial frames special? How do you determine whether a frame is inertial? Inertial to what? Problems with gravity: In equation F = GM 1M 2, the value r depends r on the frame of reference! 2 Instantaneous change in force if object s move? How does other mass find out so quickly? Why is inertial mass equal to gravitational mass? Coincidence?

2 The Equivalence Principle Consider a person in a spaceship at rest out in space, far from any source of gravity. Compare what that person feels to a person free-falling to the Earth s surface.

3 Equivalence Principle Now consider a person stationary on the surface of the Earth, and compare to a person in an accelerating rocket (far from any source of gravity).

4 Equivalence Principle Einstein s Equivalence Principle: It is impossible to tell by experiment whether you are in accelerated frame of reference or in a gravitational field. The effects of gravity is completely indistinguishable from effects of an accelerating frame

5 Equivalence Principle Consider the trajectory of a thrown ball in both frames of reference (left frame is accelerating, right frame is stationary near Earth)

6 Equivalence Principle Conversely, imagine a ball thrown sideways in both a free-falling frame (near Earth) and an elevator at rest far from any source of gravity:

7 Equivalence Principle Why do objects with different masses fall at the same rate? Easy to see why in an accelerating elevator: Floor gains on all objects at the same rate since it is the floor that is accelerating upward (the objects are not accelerating). a

8 Consequences of Equivalence Principle Mass must bend light! (Reason photons bends in accelerating elevator is identical to reason a thrown ball bends) On Earth the effect is incredibly small (too small to measure, like degrees in a 10 m laboratory). y = 1 2 gt2 = 1 2 g(l/c)2 tan θ = y L = 1 gl 2 c Rad (on Earth) On neutron star, if L=5 m, θ Rad 0.4

9 Gravitational Lens 1915: Einstein predicted observable effect due to Sun: Proper treatment requires integration over lightpath, accounting for changing values of g. Angle of deflection θ = 4GM Rc 2 =1.7 Confirmed during solar eclipse by Eddington (1919)

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12 Gravitational Redshift Consider accelerating elevator: Detector s velocity when receiving signal is different than source s velocity when emitting signal. v a t ah/c Doppler shift causes observed wavelength to be longer than emitted wavelength Detector a v 2 Detector v 1 a λ λ v c ah c 2 v 1 source v 2 source

13 Gravitational Redshift Same thing should occur on Earth (due to the equivalence principle). Detector a distance h above Earth s surface should see a gravitational redshift: Detector λ λ gh c 2 source

14 Gravitational Time Dilation The phenomena of gravitational redshift is attributed to time dilation Which clock ticks more quickly? a) Clock #1 b) Clock #2 c) They tick at the same rate! Clock #1 Clock #2

15 Geometry The basic idea of general relativity is that matter effect s the geometry of space-time. Matter does not produce a force on an object. Matter causes space-time to be curved (or warped) such that straight trajectories (resulting from a coasting object) appear curved when projected on a flat canvas. What do we mean by straight? For 3D geometry in a Euclidean ( flat ) coordinate system, a straight line is the trajectory in which the distance between two points is a minimum l = x 2 + y 2 + z 2

16 Flat Space-Time (SR) For GR, we have to consider the curvature of 4-dimensional space-time. Minkowski space-time metric is flat (similar to Euclidean 3d) s 2 =(c t) 2 ( x 2 + y 2 + z 2 ) s 2 =(c t) 2 ( r 2 + r 2 θ 2 + r 2 sin 2 θ φ 2 ) s = c2 1 ds = dt 2 dl 2 = (v/c)2 cdt ct (x 1,ct 1 ) (x 2,ct 2 ) x Δs is frame invarient: Proper time. In frame where v=0, An object coasting has a trajectory in which the overall space-time interval is a minimum. This gives s = c τ x(τ),y(τ),z(τ), and t(τ)

17 Curved Space-Time Mass warps space and time. The metric must contain curvature terms: 3 3 s 2 = g ij x i x j = g ij x i x j i=0 j=0 s 2 = g 00 t 2 + g 11 x 2 + g 22 y 2 + g 33 z 2 + g 12 x y + g 13 x z + g 23 y z + g 01 t x For flat spacetime, g 00 = c 2, g 11 = g 22 = g 33 = 1 Values (or functions) of the gs different than these values represent curved space-time An object coasting has a trajectory in which the overall spacetime interval is a minimum. This gives (after application of calculus of variations) x(τ),y(τ),z(τ), and t(τ)

18 Curved Surface What is the shortest trajectory to fly from Denver to London? If you look at trajectory on a Euclidean map (ignoring curvature of Earth), trajectory is curved!

19 Schwarzschild Metric Schwarzschild found a solution to Einstein s field equations for outside a spherical object of mass M and radius R. s 2 = R rc 2 c 2 t 2 r2 rc 2 r 2 ( θ 2 +sin 2 θ φ 2 ) r r,θ,φ, and t are coordinates a far-away observer (ignorant of curvature) uses. All observers agree on the value of the spacetime interval, s

20 Gravitational Time Dilation Revisited s 2 = rc 2 Consider a stationary person a distance r from center of spherical object. This person measures his proper time, Δτ =Δs/c. If the observer doesn t move, then Thus the reading of the far-away clock, Δt, differs from the reading of the person s clock Δτ via τ = Gravitational redshift: c 2 t 2 r2 rc 2 r 2 ( θ 2 +sin 2 θ φ 2 ) rc 2 s = c τ = t ν 1 t rc 2 c t ν = ν 0 rc 2 λ 1 = λ 0 rc 2

21 Gravitational Length Expansion Now consider two points located at r 1 and r 2 (in line with the center of the spherical mass). What is the distance between the two points? Far-away ignorant observer would say L=Δr (=r 2 -r 1 ). But Schwarzschild metric now gives: r s = rc 2 Actual space is stretched due to mass. = L

22 Time Delay of light travel Consider motion of photon moving from position r 1 to r 2 M R r 1 r 2 Spacetime interval for photon is zero: ds 2 =0=c 2 dt 2 c 2 dr2 r rc 2 t = t = r 2 r 1 c dt = 1 c r2 r 1 + 2GM c 3 dr rc 2 r2 2GM c ln 2 r c 2 light takes longer (according to our clock) due to both space stretching and time dilation

23 Trajectories Near Compact Object Consider purely radial motion ds 2 = rc 2 c 2 dt 2 dr2 rc 2 s = rc 2 c 2 dt 2 dr2 rc 2 = rc 2 c 2 ( t/ τ) 2 1 rc 2 ( r/ τ) 2 dτ Using calculus of variations to find an extremum, the functions r(τ) and t(τ) can be determined [and thus ] r(t) Mike Guidry, Univ. Tenn,Knoxville

24 Experimental Tests of Time Dilation 1959 Pound-Rebka Esperiment: Gravitational redshift measured at Harvard (a 22 m building) using Iron-57 gamma rays. Change in frequency is ν ν Echo delay off Venus (Shapiro Delay); pulse is 0.1 ms delayed in the 20 minute round trip) (1966) GPS devices require the use of GR for sub 1-meter positioning accuracy ν ν Synchronized clocks, with one placed on airplane or satellite, don t remain synchronized. Gravitational redshift detected on neutron stars and white dwarfs.